On Expansion of Regularity of Nonlinear Evolution Equations by Means of Dilation Symmetry

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On Expansion of Regularity of Nonlinear Evolution Equations by Means of Dilation Symmetry

Andrey Polyakov

To cite this version:

Andrey Polyakov. On Expansion of Regularity of Nonlinear Evolution Equations by Means of Dilation

Symmetry. 2019. �hal-02093984v2�

Noname manuscript No.

(will be inserted by the editor)

On Expansion of Regularity of Nonlinear Evolution Equations by Means of Dilation Symmetry

Andrey Polyakov

Received: date / Accepted: date

Abstract The paper present a dilation symmetry based approach to expan- sion of regularity of nonlinear evolution equations. In particular, it is shown that a symmetry of an operator, which describes a right-hand side of a non- linear evolution equation, is inherited by solutions of this equation. In the case of dilation symmetry, the latter implies that global-in-time existence of solu- tions for small initial data always imply global-in-time existence of solutions for large initial data. As an example, we consider the problem of expansion of regularity of the Navier-Stokes equations (in R

) accepting that the existence of global-in-time solutions for small initial data is already proven.

Keywords Dilation Symmetry · Nonlinear Evolution Equations

1 Introduction

According to the classical concept of homogeneity introduced by Leonhard Euler in 18th century, homogeneity is a sort of symmetry of an object (e.g.

a function or a set) with respect to a group of transformations knows today as dilations. For example, a function f in R

is homogeneous in the classical (standard) sense if it is symmetric with respect to a uniform dilation of an argument, i.e. there exists ν ∈ R such that

f (e

u) = e

f (u), u ∈ R

, s ∈ R .

Homogeneity of a function is inherited by other objects induced by this func- tion. For example, the Euler’s Homogeneous Function Theorem implies that

A. Polyakov

Inria Lille, Univ. Lille, CNRS, UMR 9189 - CRIStAL, (F-59000 Lille, France), Tel.: +33-359577802

E-mail: andrey.polykov@inria.fr

any derivative of f is homogeneous too. Similarly, if u(·) : [0, +∞) → R

is a classical solution of

du

dt = f (u), t > 0

with the initial condition u(0) = u

then u

(t) := e

u(e

t) is defined on [0, +∞), then, due to symmetry, we derive

= e

f (u(e

t)) = f (u

(t)), t > 0, i.e. u

is a classical solution of the same differential equation with the initial condition u

(0) = e

u

, where s ∈ R .

Let ∃ε > 0 such that a classical solution of the differential equation exists on [0, +∞) for any initial value u(0) = u

∈ B

:= {u ∈ R

: kuk < ε}.

Hence, exploiting the symmetry we derive existence of a classical solution for any initial condition u(0) = u

∈ S

s∈R

e

B

= R

.

For non-linear evolution system, it may be simpler to prove existence and uniqueness of a regular solution for small initial data. In this paper we show that the dilation symmetry can be utilized for global expansion of regularity non-linear evolution equations in a Banach space B provided that a dilation group is properly introduced in B . As an example, we consider the problem of expansion of regularity of the Navier-Stokes equations (in R

) accepting that the existence of global-in-time solutions for small initial data is already proven.

In particular, we present a necessary and sufficient conditions for expansion of regularity of Navier-Stokes equation by means of dilation symmetry.

Mainly, the standard notation is utilized through the paper, e.g. R is the field of real numbers; L

((0, T )× R

, R ) denotes the space of locally integrable functions (0, T ) × R

→ R ; L

( R

, R

), 1 ≤ p ≤ +∞ is a Lebesgue space of function R

→ R

with the norm k · k

; C

((0, T ) × R

, R

) is a space of smooth functions (0, T ) × R

→ R

with compact support and C

([0, T ) × R

, R

) is a space of smooth functions which vanish at infinity, where 0 <

T ≤ ∞. For composition of operators A, B we also use the notation A ◦ B.

Let L

( R

, R

) denotes the following normed vector space of functions R

→ R

L

( R

, R

) := {u : kuk

< +∞} , µ ∈ R kuk

:=

Z

Rⁿ

|x|

|u(x)|

dx

, 0 < p < ∞ kuk

:= ess sup(|x|

u(x)), p = ∞,

which can be treated as a wighted L

. The notation

∈ and

= is utilized in order to indicate that an inclusion or identity is fulfilled almost everywhere on a domain.

2 Symmetry of nonlinear operators

2.1 Dilation group

Let X be a (linear) vector space and {d(s)}

be a family of operators d(s) :

X → X . If

• d(0) = I, where I is an identity operator (i.e. Iz = z for all z ∈ X );

• d(t + s)z = (d(t) ◦ d(s))z = (d(s) ◦ d(t))z for all t, s ∈ R , z ∈ X

then, by definition, d is a group. Using the group properties for t = −s we derive (d(−s) ◦ d(s))z = (d(s) ◦ d(−s))z = z, the operator d(s) is invertible and [d(s)]

= d(−s). Moreover, d(s) maps X onto X for any s ∈ R . Indeed, suppose the contrary, i.e. ∃z

∈ X and such s

∈ R such that z

∈ / d(s

) X . Since u

:= d(−s

)z

∈ X then z

= d(s)d(−s

)z

= d(s)u

∈ d(s) X .

Definition 1 A group d of operators on a normed vector space X is said to be a dilation group (or simply dilation) on X if d(s) 0 = 0 for any s ∈ R and the following limit property holds

lim inf

s→−∞

kd(s)zk = 0 and lim sup

s→+∞

kd(s)zk =∞ for z 6= 0 .

The limit property given above specifies a group being a dilation in an abstract space. We refer the reader to [2] for more details about topological characterization of dilations.

Example 1 Let us recall a few well-know dilation groups in R

: 1) Uniform dilation (L. Euler, 18th century):

d(s) = e

I, s ∈ R where I is the identity matrix R

.

2) Weighted dilation [14]:

d(s) =









e

0 ... 0 0 e

... 0 ... ... ... ...

0 ... ... e







 ,

where r

> 0, i = 1, 2, ..., n.

3) Geometric dilation (see e.g. [4], [12], [3]) is a flow generated by an Euler vector field

.

The uniform dilation d(s) = e

I ∈ B , s ∈ R is the simplest example of a dilation in any normed vector space. Let us consider a few other examples.

Example 2 Let B be a space of bounded uniformly continuous functions R

→ R

with the supremum norm. A dilation group d in B can be defined as follows

(d(s)z)(x) = e

z(x + βs),

1 A C¹ vector fieldν : Rⁿ → Rⁿ is called Euler if it is complete and −ν is globally asymptotically stable.

where s ∈ R is the group parameter, z ∈ X , x ∈ R and α > 0 and β ∈ R are a constant parameters. Indeed, d(s)z ∈ X if z ∈ B , s ∈ R and for v = d(s)z we have

(d(t)◦d(s)z)(x) = (d(t)v)(x) = e

v(x+βt) =e

e

z(x+βs+βt) = (d(s+t)z)(x).

The limit property also holds since kd(s)zk = sup

x∈Rⁿ

|e

z(x + s)| = e

sup

x∈Rⁿ

|z(x + βs)| = e

kzk.

The next lemma introduces the most common dilation in functional spaces.

Lemma 1 The operator d(s) given by

(d(s)z)(x) = e

z(e

x), (1) where s ∈ R , z is a function R

→ R

, x ∈ R

and α, β ∈ R are constant parameters, is

– a linear bounded invertible operator on L

( R

, R

),

kd(s)zk

= e

kzk

, z ∈ L

( R

, R

), s ∈ R , – a linear bounded invertible operator on L

( R

, R

),

kd(s)zk

= e

kzk

, z ∈ L

( R

, R

), s ∈ R , where 0 < p ≤ ∞. The inverse operator is given by [d(s)]

= d(−s).

Proof Notice that L

= L

.

Let 1 ≤ p < ∞. If z ∈ L

( R

, R

) then Z

Rⁿ

|x|

|z(x)|

dx < +∞

and Z

Rⁿ

|x|

|z(x)|

dx = e

Z

Rⁿ

|e

x|

|z(e

x)|

dx = e

Z

Rⁿ

|x|

(d(s)z)(x)|

dx < +∞.

Since e

> 0 for any α, β, p, s ∈ R then d(s)z ∈ L

( R

, R

) for any s ∈ R . Obviously, d(s) is a linear operator on L

, i.e. d(s)(µ

z

+ µz

) = µ

d(s)z

+ µ

d(s)z

, for any µ

, µ

∈ R and z

, z

∈ L

( R

, R

). Moreover, the latter identities imply that

kd(s)zk

=e

kzk

, kzk

:=

Z

Rⁿ

|x|

|z(x)|

dx

.

Hence, the operator d(s) : L

( R

, R

) → L

( R

, R

) is bounded for any s ∈ R .

Let p = ∞. If z ∈ L

( R

, R

) then

ess sup|z(x)| = ess sup(|e

x|

|z(e

x)|) < +∞

for any β, s, µ ∈ R and kd(s)zk

= e

kzk

for any s ∈ R . Therefore, d(s) is also a linear bounded operator on z ∈ L

( R

, R

).

Obviously, (d(s) ◦ d(−s))z = (d(−s) ◦ d(s))z for any z : R

→ R

and any s ∈ R and we derive [d(s)]

= d(−s).

2.2 d-homogeneous operators

In this section we introduce a notion of d-homogeneous (symmetric with re- spect to a group d) operators in a vector space X and present a couple of examples.

Definition 2 An operator F : D(F ) ⊂ X → X is said to be d- homogeneous of degree µ ∈ R if d(s)D(F) ⊂ D(F ) for any s ∈ R and

(F ◦ d(s))u = e

(d(s) ◦ F )u for s ∈ R , u ∈ D(F ), (2) where d is a group of invertible operators on X .

A lot of examples of d-homogeneous vector-field for X = R

can be found in control literature (see e.g. [14], [12], [3] and references therein). For example, the vector function

(x

, x

) →

x

+ x

, |x

| sin

x

− x

|x

| + x

, (x

, x

) ∈ R

is d-homogeneous of degree 0 with

d(s)(x

, x

) → (e

x

, e

x

), s ∈ R .

All linear and lot of nonlinear models of mathematical physics are d-homogeneous under a proper selection of a dilation group (see e.g. [11]).

Notice that, the identity (2) can always be understood in the weak sense.

For shortness we omit R

in the notations for R

Rⁿ

, L

( R

, R

),L

( R

, R

) and C

( R

, R

) in the examples below.

Example 3 (d-homogeneity of the Laplace operator) Let us consider the Laplace operator

∆ : D(∆) ⊂ L

→ L

,

with the domain D(A) =

u ∈L

: ∃f ∈ L

such that Z

u · ∆φ = Z

f · φ, ∀φ∈ C

.

Let us show that ∆ is d-homogeneous of degree 2β provided that the dilation d is given by (1).

By Lemma 1, d is a group of linear invertible operators on C

and, con- sequently (see the beginning of Section 2), d(s) maps C

onto C

. Notice that if φ ∈ C

then, obviously,

(∆ ◦ d(s))φ)(x) =e

(∆φ)(e

x) = e

((d(s)◦ ∆)φ)(x), s∈ R , x∈ R

. In other words, the Laplace operator is d-homogeneous as operator C

→ C

. Since C

is dense in L

then it is d-homogeneous as an operator L

→ L

. Let us prove this claim more rigorously.

Let u ∈ D(∆) and ∆u = f ∈ L

(in the weak sense). Since d(s)f ∈ L

then using the change-of-variable theorem (see e.g. [6]) in the Lebesgue integral we derive

e

Z

(d(s)f )·φ =e

Z

f (e

x)·φ(x)dx =e

Z

f (x)·φ(e

x)dx =

e

Z

f · φ ˜ = e

Z

u· ∆ φ ˜ = e

Z

u· (∆ ◦ d(−s))φ =

e

Z

u(x) · ∆φ(e

x)dx =e

Z

u(e

x) · ∆φ(x)dx = Z

(d(s)u) · ∆φ, where ˜ φ =d(−s)φ ∈ C

. Hence, d(s)u ∈ D(∆) and (∆ ◦d(s))u = e

d(s)f = e

(d(s) ◦ ∆)u (in the weak sense) for any s ∈ R , u ∈ D(∆).

3 Symmetry of Evolution Equations

3.1 Linear evolution equation

The dilation symmetry of an operator is inherited by other objects induced (generated) by this operator.

Lemma 2 Let a linear closed densely defined operator A : D(A) ⊂ B → B generate a strongly continuous semigroup Φ := {Φ(t)}

of linear bounded operators on B and d be a group of linear bounded invertible operators on B . If the operator A is d-homogeneous of degree µ then

Φ(t) ◦ d(s) = d(s) ◦ Φ(e

t), ∀t ≥ 0, ∀s ∈ R . (3)

Proof Since Φ is generated by A then Φ(e

t)u ∈ D(A) for any u ∈ D(A) (see e.g. [8, page 5]).

Let s ∈ R and u ∈ D(A) be selected arbitrary. Since the operator A is d- homogeneous then D(A) is invariant with respect to the transformation d(s), i.e. d(s)z ∈ D(A), ∀z ∈ D(A), and, consequently,

y

(t) := (Φ(t)◦d(s))u ∈ D(A), and y

(t) := (d(s)◦ Φ(e

t))u ∈ D(A), t ≥ 0.

Being generated by A the semigroup Φ satisfy (see e.g. [8, page 5]) d

dt Φ(t)z = (A ◦ Φ(t))z = (Φ(t) ◦ A)z, ∀t > 0, ∀z ∈ D(A).

Taking into account that A is d-homogeneous of degree µ and d(s) is a linear bounded operator on B , we derive

d

dt

y

(t) = e

(d(s) ◦ A ◦ Φ(e

t))u= (A ◦ d(s) ◦ Φ(e

t))u= Ay

(t), ∀t > 0.

On the other hand, we have

d

dt

y

(t) = (A ◦ Φ(t) ◦ d(s))u = Ay

(t), ∀t > 0.

Since y

(0) = y

(0) = d(s)u then due to uniqueness of the semigroup Φ generated by A (see [8, page 6]) we derive y

(t) = y

(t) for all t ≥ 0 and

(Φ(t) ◦ d(s))u = (d(s) ◦ Φ(e

t))u, ∀t ≥ 0, ∀u ∈ D(A).

Since Φ(t) ◦ d(s) and d(s) ◦ Φ(e

t) are bounded linear operators and D(A) is dense in B then the latter identity holds for all u ∈ B and all t ≥ 0.

It is well-known (see e.g. [8, page 100]) that u(t, u

) = Φ(t)u

, t ≥ 0 is a unique solution of the linear evolution equation

du dt = Au

with the initial condition u(0) = u

∈ B . The latter lemma, obviously, proves the symmetry of these solutions:

u(t, d(s)u

) = d(s)u(e

t, u

), s ∈ R

Below we prove this result for non-linear operators too.

3.2 Nonlinear evolution equation

Let us consider the non-linear evolution system du

dt = Au + Gu, t > 0, (4)

where a densely defined closed linear operator A : D(A) ⊂ B → B

generates a strongly continuous semigroup Φ of linear bounded operators on B , and

G : D(G) ⊂ B → B is a possibly nonlinear operator.

Definition 3 A continuous function u : [0, T ) → B is said to be – a mild solution of the system (4) if Gu ∈ L

((0, T ), B ) and

u(t) = Φ(t)u(0) + Z

0

(Φ(t − τ ) ◦ G)u(τ ) dτ, t ∈ (0, T ); (5) – a strong solution of the evolution equation (4) if u ∈ C([0, T ), B ), u is differentiable almost everywhere on (0, T ),

, Gu ∈ L

((0, T ), B ) and (4) is satisfied almost everywhere on (0, T );

– a classical solution of the evolution equation (4) if u ∈ C([0, T ), B ),

du

dt

∈ C((0, T ), B ), u(t) ∈ D(A) ∩ D(G) for all t ∈ (0, T ) and (4) is satisfied on (0, T ).

The latter integral is understood in the sense of Bochner ([13, page 132]).

Theorem 1 Let d be a group of linear bounded invertible operators on B and let A and G be d-homogeneous operators of a degree µ ∈ R .

If u : [0, T ) → B is a mild (or strong) solution of the evolution equation (4) and

u(t)

∈ D(G), t ∈ (0, T ), then the function u

: [0, e

T) → B defined as

u

(t) = d(s)u(e

t), t ∈ [0, e

T )

is also a mild (resp. strong) solution of the evolution equation (4) and

u

(t)

∈ D(G), t ∈ (0, e

T ),

for any s ∈ R .

Moreover, the claim remains true for classical solutions and the above inclusions hold everywhere on (0, T ) and (0, e

T), respectively.

Proof If u(t) ∈ D(G) then due to d-homogeneity of the operator G we have d(s)D(G) ⊂ D(G) and u

(e

t) ∈ D(G).

1) The case of mild solutions.

Since the d-homogeneous operator A generates a strongly continuous semi- group Φ, then according Lemma 2 we have

Φ(t) ◦ d(s) = d(s) ◦ Φ(e

t), ∀t ≥ 0, ∀s ∈ R . It is well known [13, page 134] that K R

0

ξ(s)ds = R

0

Kξ(s)ds for any bounded linear operator K : B → B and any Bochner integrable function ξ ∈ L

((0, T ), B ).

Hence, using G(u) ∈ L

((0, T ), B ) we derive d(s)

Z

G(u(τ))dτ = Z

0

d(s)G(u(τ))dτ = e

Z

0

G(d(s)u(τ))dτ =

Z

G(d(s)u(e

τ ))dτ = Z

0

G(u

(τ))dτ, i.e. G(u

) ∈ L

((0, e

T ), B ). Similarly, we derive

d(s)u(e

t) = (d(s) ◦ Φ(e

t))u(0) + Z

0

(d(s) ◦ Φ(e

t − τ ) ◦ G)u(τ ) dτ =

(Φ(t) ◦ d(s))u(0) + Z

0

(d(s) ◦ Φ(e

t − τ) ◦ G)u(τ) dτ =

(Φ(t) ◦ d(s))u(0) + e

Z

0

(d(s) ◦ Φ(e

(t − σ)) ◦ G)u(e

σ)) dσ =

(Φ(t) ◦ d(s))u(0) + e

Z

0

(Φ(t − σ) ◦ d(s) ◦ G)u(e

σ)) dσ =

(Φ(t) ◦ d(s))u(0) +

t

Z

0

(Φ(t − σ) ◦ G)(d(s)u(e

σ)) dσ,

where the linearity of the operator Φ(t − σ) and the d-homogeneity of the operator G are utilized on the last step. Therefore, we have shown that

u

(t) = Φ(t)u

(0) +

t

Z

0

(Φ(t − σ) ◦ G)u

(σ) dσ,

i.e. u

is a mild solution of (4).

2) The case of strong and classical solutions. Let u be a strong solution.

Since u(t) is differentiable almost everywhere and d(s) is a linear bounded operator then for τ = e

t we have

du_s(t) dt

a.e.

= e

d(s)

= e

d(s)(Au(τ)+G(u(τ))

= Au

(t)+G(u

(t)).

In the case of the classical solution the latter identity holds everywhere.

Moreover, if u(t) ∈ D(A) then due to d-homogeneity of the operator A we always have u

(t) ∈ D(A). Finally, taking into account that the operator d(s) is bounded we conclude u

∈ C([0, T ), B ) provided that u ∈ C([0, T ), B and

d

dt

u

∈ C((0, T ), B ) provided that

u ∈ C((0, T ), B , i.e. u

is also a classical solution.

Remark 1 The proven symmetry of solutions of the evolution equation (4) allows us to guarantee that a local result holds globally. For example, let us consider the initial value problem u(0) = u

for (4). If D(G) = B then an existence of a solution for any u

∈ B(r) implies existence of solutions for all u

∈ B , where B(r) is a ball in B of the radius r > 0. Indeed, using the limit property of the dilation d we derive that for any u

∈ B there exists s ∈ R such that kd(s)u

k < r, i.e. d(s)u

∈ B (r). By Theorem 1, if u(t, d(−s)u

) is a solution of the initial value problem u(0) = d(−s)u

∈ B (r) then u

(t, u

) :=

d(s)u(e

t, d(−s)u

) is also a solution of the evolution equation (4). Since d(s)u(0, d(−s)u

) = d(s)d(−s)u

= u

then u

is a solution of the initial value problem u(0) = u

∈ B for (4).

Notice that if all solutions with u

∈ B(r) exist on [0, +∞) then all solu- tions with u

∈ B also exist on [0, +∞). Below we show possible cases when such an approach allows to expand regularity of solutions of the Navier-Stokes equations provided that it is already proven for small initial data (see e.g. [5]

and [7]).

Some other applications of a dilation symmetry in evolution equations, for example, to problems of mathematical control theory can be found in [11], [10].

3.3 Nonlinear implicit evolution equation

Let ˜ X = B × X , where B is Banach space and X is a linear (vector) space. Let us consider the nonlinear implicit evolution equation

du

dt = Au + G(u, p), 0 = Q(u, p),

t > 0, (6)

where a densely defined closed linear operator

A : D(A) ⊂ B → B

generates a strongly continuous semigroup Φ of linear bounded operators on B , and

G : D(G) ⊂ X ˜ → B and Q : D(Q) ⊂ X ˜ → X are a (possibly) nonlinear operators.

Definition 4 A pair (u, p) with u : [0, T ) → B and p : [0, T ) is said to be – a mild solution of the implicit evolution equation (6) if u ∈ C([0, T ), B ),

G(u, p) ∈ L

((0, T ), B ) and u(t) = Φ(t)u(0)+

Z

(Φ(t − τ) ◦ G)(u(τ), p(τ)) dτ, 0

= Q(u(t), p(t)),

t ∈ (0, T ); (7)

– a strong solution of the implicit evolution equation (6) if u ∈ C([0, T ), B ), u is differentiable almost everywhere on (0, T ),

du

dt

, G(u, p) ∈ L

((0, T ), B ) and (6) is satisfied almost everywhere on (0, T );

– a classical solution of the evolution equation (6) if u ∈ C([0, T ), B ),

du

dt

∈ C((0, T ), B ), u(t) ∈ D(A) and (u(t), p(t)) ∈ D(G) ∩ D(Q) for t ∈ (0, T ) and (6) is satisfied on (0, T ).

A symmetry of solutions of the implicit evolution equation (6) is also pre- served provided that the operator, which defines its right-side is d-homogeneous.

Theorem 2 Let a group d of invertible operators on X ˜ be defined as follows

d(s)(u, p) = (d

(s)u, d

(s)p), s ∈ R , u ∈ B , p ∈ X ,

where d

is a group of linear bounded invertible operators on B , d

is a group of invertible operators on X such that

d

(s) 0 = 0, s ∈ R . Let us denote F := (G, Q) and D(F) := D(G) ∩ D(Q),

F : D(F ) ⊂ X ˜ → X ˜ .

Let A be a d

-homogeneous operator of a degree µ ∈ R and F be d- homogeneous operator of the same degree µ.

If the pair (u, p) with u : [0, T ) → B and p : [0, T ) → X is a mild (or strong) solution of the implicit evolution equation (6) such that

(u(t), p(t))

∈ D(F), t ∈ (0, T ),

then the pair (u

, p

) with u

: [0, e

T ) → B and p : [0, T ) → X defined as follows

(u

(t), p

(t)) := d(s)(u(e

t), p(e

t)) with t ∈ [0, e

T ) is also a mild (resp. strong) solution of the evolution equation (4) and

(u

(t), p

(t))

∈ D(F ), t ∈ (0, e

T ) for any s ∈ R .

Moreover, the claim remains true for classical solutions and the above inclusions hold everywhere on (0, T ) and (0, e

T), respectively.

Proof Since the operator F is d homogeneous then d(s)D(F ) ⊂ D(F ) and

G(d

(s)u, d

(s)p) = e

(d

(s) ◦ G)(u, p), ∀(u, p) ∈ D(F ), ∀s ∈ R , Q(d

(s)u, d

(s)p) = e

(d

(s) ◦ Q)(u, p), ∀(u, p) ∈ D(F ), ∀s ∈ R . Hence, if (u(t), p(t)) ∈ D(F) we conclude (u

(e

t), p

(e

t)) ∈ D(F ).

1) The case of mild solutions.

Since the d

-homogeneous operator A generates a strongly continuous semigroup Φ, then according Lemma 2 we have

Φ(t)d

(s) = d

(s)Φ(e

t), ∀t ≥ 0, ∀s ∈ R . It is well known [13, page 134] that K R

0

ξ(s)ds = R

0

Kξ(s)ds for any bounded linear operator K : B → B and any Bochner integrable function ξ. Hence, using G(u, p) ∈ L

((0, T ), B ) we derive

d

(s) Z

0

G(u(τ), p(τ))dτ = Z

0

d

(s)G(u(τ ), p(τ ))dτ =

e

Z

0

G(d

(s)u(τ), d

(s)p(τ))dτ = Z

0

G(d

(s)u(e

τ), d

(s)p(e

τ))dτ

= Z

0

G(u

(τ), p

(τ ))dτ, i.e. G(u

, p

) ∈ L

((0, e

T ), B ). Similarly, we derive d

(s)u(e

t) = (d

(s)◦Φ(e

t))u(0)+d

(s)

Z

(Φ(e

t−τ)◦G)(u(τ), p(τ)) dτ

= (Φ(t) ◦ d

(s))u(0) + Z

0

(d

(s) ◦ Φ(e

t − τ) ◦ G)(u(τ ), p(τ)) dτ =

(Φ(t) ◦ d

(s))u(0) + e

Z

0

(d

(s) ◦ Φ(e

(t − σ)) ◦ G)(u(e

σ), p(e

σ)) dσ =

(Φ(t) ◦ d

(s))u(0) + e

Z

0

(Φ(t − σ) ◦ d

(s) ◦ G)(u(e

σ), p(e

σ)) dσ =

(Φ(t) ◦ d

(s))u(0) +

t

Z

0

(Φ(t − σ) ◦ G)(d

(s)u(e

σ), d

(s)p(e

σ)) dσ, where the linearity of Φ(t −σ) and a homogeneity of the operator G is utilized on the last step. Therefore, we have shown that

u

(t) = Φ(t)u

(0) +

t

Z

0

Φ(t − σ)g(u

(σ), p

(σ)) dσ.

Finally, since (u, p) is a mild solution on [0, T ) then Q(u(t), p(t))

= 0, ∀t ∈ [0, T ) and using the identity d

(s) 0 = 0 we derive

Q(u

(τ), u

(τ)) = Q((d

(s)u)(τ), (d

(s)p)(τ)) = e

(d

(s) ◦ Q)(u(e

τ), p(e

τ))

= 0, τ ∈ [0, e

T ).

2) The case of strong and classical solutions. Let (u, p) be a strong solution.

Since u(t) is differentiable almost everywhere and d(s) is a linear bounded operator then for τ = e

t we have

dus(t) dt

a.e.

= e

d(s)

= e

d(s)(Au(τ)+ G(u(τ ), p(τ ))

= Au

(t)+ G(u

(t), p

(t)).

In the case of the classical solution the latter identity holds everywhere. More- over, if u(t) ∈ D(A) then due to d-homogeneity of the operator A we always have u

(t) ∈ D(A), i.e. u

is a classical solution.

4 Example: On necessary and sufficient conditions of global existence solutions of Navier-Stokes Equations in R

The Navier-Stokes equations,

∂

u = ν∆u − (u · ∇)u − ∇p, 0 = divu

where u denotes the velocity of a fluid, p denotes the scalar pressure and

ν > 0 denotes viscosity of the fluid, is the the classical model of the flow of

an incompressible viscous fluid. Without loss of generality ([7, page 4])we can assume ν = 1.

Below for shortness we omit R

in the notations of R

Rⁿ

, L

, C

and C

spaces if the context is clear.

The classical idea of analysis is to prove, initially, an existence and regu- larity of weak solutions and next to show that any weak solution is smooth.

For Navier-Stokes equations this scheme has been realized by Jean Leray in 1938 (see, e.g. [5] and the recent review [7]). We follow the same way having in mind d-homogeneity (dilation symmetry) of Navier-Stokes equations (see e.g.

[1, formula (1.5)]), which will guarantee a global expansion of all local results.

4.1 Dilation Symmetry of Navier-Stokes equation in R

Let V be a set of the so-called weakly divergence free velocity fields:

V :=

u ∈ L

: Z

u · ∇φ = 0, ∀φ ∈ C

.

Due to this the Navier-Stokes equations can be equivalently represented in the form of the d-homogeneous implicit evolution equation (6) with the operators

A = ∆, G(u, p) = −∇p − (u · ∇)u, Q(u, p) = divu having the domains D(Q) = V × L

( R

, R ),

D(A) =

u ∈L

: ∃f ∈ L

such that Z

u · ∆φ = Z

f · φ, ∀φ∈ C

,

D(G) =

(u, p)∈ V×L

: ∃f ∈ L

, Z

u·(u·∇)φ−pdivφ+f ·φ = 0, ∀φ∈ C

.

Lemma 3 Let d(s) : ˜ X → X ˜ with X ˜ := L

× L

( R

, R ) be defined as d(s)(u, p) = (d

(s)u, d

(s)p), s ∈ R , u ∈ L

, p ∈ L

( R

, R ), where d

is a group of linear bounded invertible operators on L

,

(d

(s)u)(x) = e

u(e

x), u ∈ L

, x ∈ R

,

and d

is a group of linear invertible operators on L

( R

, R ) given by (d

(s)p)(x) = e

p(e

x) p ∈ L

, x ∈ R

.

Then the operator F : D(F ) ⊂ X ˜ → X ˜ with D(F) = D(G) ∩ D(Q) and

F = (G, Q) is d-homogeneous of degree 2.

Proof To complete the proof we must show that

G(d

(s)u, d

(s)p) = e

(d

(s) ◦ G)(u, p), ∀(u, p) ∈ D(G), ∀s ∈ R , Q(d

(s)u, d

(s)p) = e

(d

(s) ◦ Q)(u, p), ∀(u, p) ∈ D(Q), ∀s ∈ R . and the domain D(F) is invariant with respect to d(s), ∀s ∈ R .

1) According to the definition of Q, the identity Q(u, p) = 0 ∈ L

(in the weak sense) means

Z

u · ∇φ = 0, ∀φ ∈ C

.

and using a change-of-variable theorem in Lebesque integral we derive e

Z

kuk

u(e

x) · (∇φ)(e

x) = 0, ∀φ ∈ C

.

By Lemma 1 we have d

(s)u ∈ L

for any s ∈ R . Taking into account

∇(d

(s)φ)(x) = e

(∇φ)(e

x) we obtain e

Z

(d

(s)u) · (∇d

(s)φ) = 0, ∀φ ∈ C

. Since d

(s) maps C

onto C

then d

(s)V ⊂ V for any s ∈ R and

(Q ◦ d(s))(u, p) = 0 = e

d

(s) 0 = e

(d

(s) ◦ Q)(u, p) 1 for all s ∈ R and, at least, on D(Q).

2) The identity G(u, p) = f ∈ L

(in the weak sense) means Z

u·(u·∇)φ − p divφ+f · φ = 0, ∀φ∈ C

.

If p ∈ L

and u ∈ L

then from Lemma 1 we derive d

(s)u ∈ L

and d

(s)p ∈ L

.

Using the change-of-variable theorem in the Lebesgue integral we derive e

Z

u(e

x) ·(u(e

x)·(∇φ)(e

x)) + p(e

x) (divφ)(e

x)+ f (e

x) · φ(e

x) dx = 0.

or, equivalently, Z

(d

(s)u)·(d

(s)u·∇)d

(s)φ+(d

(s)p) div(d

(s)φ)+e

(d

(s)f )·(d

(s)φ) = 0, for all φ ∈ C

. Since d

(s) maps C

onto C

then we conclude that d(s)D(G) ⊂ D(G) and the identity

G(d

(s)u, d

(s)p) = e

d

(s)f = e

d

(s)G(d

(s)u, d

(s)p) holds for any s ∈ R and, at least, on D(G).

Therefore, F is, indeed, d-homogeneous of degree 2.

The d-homogeneity of the Laplace operator ∆ is studied in Example 3.

Therefore, the Navier-Stokes equation satisfy all conditions required for appli-

cation of Theorem 2. In other words, if it has mild, strong or classical solutions

defined on [0, +∞) for small initial values then it has, respectively, mild, strong

or classical solutions defined on [0, +∞) for large initial values.

4.2 Necessary and sufficient conditions for global existence of solutions in R

Corollary 1 Let q

, q

∈ [1, ∞] and r

, r

, µ

, µ

∈ R and

r

(1 − µ

− n/q

) + r

(1 − µ

− n/q

) 6= 0. (8) A mild (strong or classical) solution of the Navier-Stokes equations with arbitrary the initial data u(0) = u

∈ V ∩ L

∩ L

exists on [0, +∞) if and only if there exist ε > 0 such that for any

u

∈ {u ∈ V ∩ L

1

∩ L

2

: kuk

1,µ₁

kuk

2,µ₂

< ε}

a mild (resp. strong or classical) solution (u, p) with the initial data u(0) = u

exists on [0, +∞).

Proof Let d

be defined as in the proof of Lemma 3. Assume that for any u

∈ {u ∈ V ∩ L

∩ L

: kuk

kuk

< ε} there exists a global in time mild (strong or classical) solution with u(0) = u

and let us show that for any u

∈ V ∩ L

∩ L

: ku

k

ku

k

≥ ε the Navier-Stokes equations also have a mild( resp. strong or classical) solution on [0, +∞).

In the proof of Lemma 3 we have shown that d

(s)u

∈ V for any s ∈ R , by Lemma 1 and d

(s)u

∈ L

and for d

(s)u

∈ L

any s ∈ R . By Lemma 1 we also derive

kd

(s)u

k

= e

ku

k

, and

kd

(s)u

k

= e

ku

k

and

kd

(s)u

k

1,µ₁

kd

(s)u

k

2,µ₂

= e

ku

k

1,µ₁

ku

k

2,µ₂

. Since by assumption r

(1 − µ

− n/q

) + r

(1 − µ

− n/q

) 6= 0 then for any u

∈ V ∩ L

∩ L

there exists s

∈ R such that

kd

(s)u

k

kd

(s)u

k

< ε.

Hence, if a strong solution (u, p) with u(0) = d

(s

)u

exists on [0, +∞) then by Theorem 2 the pair (˜ u, p) given by ˜

˜

u(t, x) = e

u(e

t, e

x), p(t, x) =e ˜

p(e

t, e

x), t ∈ [0, +∞), x ∈ R

is also a strong solution of the Navier-Stokes equation. Since u(0) = d

(s

)u

means that

u(0, x) = e

u

(e

x), x ∈ R

then

˜

u(0, x) = e

u(0, e

x) = u

(x),

i.e. ˜ u(0) = u

∈ V ∩ L

∩ L

and the proof is complete.

Notice that taking µ

= µ

= 0 we derive the usual L

and L

spaces in the latter corollary.

Let us consider also the well-known weak form (see e.g. [5] and the recent review [7]) of Navier-Stokes equations

Z

u(0) · φ(0) +

T

Z

0

Z

u · (φ

+ ∆φ) + p divφ =

T

Z

0

Z

u · (u · ∇)φ, ∀φ ∈ C

(9)

with u(t) ∈ V for t ∈ (0, T ).

Definition 5 [7, Definition 3.7] A pair (u, p) is said to be a solution of the Navier-Stokes equations on [0, T ) if p ∈ L

((0, T ) × R

, R )), u(t) ∈ V , t ∈ (0, T )

u ∈ C([0, T ), L

) ∩ C((0, T ), L

), ku(t)k

is bounded as t → 0

and (9) is satisfied.

According to [7] the considered solution can be treated as a mild solutions obtained using the so-called Oseen kernel.

Let us mention the following properties of the solutions introduced by Definition 5 (proven for n = 3):

– Global-in-time existence for small initial data [7, Corollary 3.13 and Lemma 3.10]

There exist ε > 0 and C > 0 such that for any

u

∈ {u ∈ V : kuk

kuk

< ε} (10) or

u

∈ {u ∈ V : kuk

k∇uk

< ε} (11) or

u

∈ {u ∈ V : kuk

kuk

< ε}, q > 3 (12) a strong solution (u, p) with the initial data u(0) = u

exists on [0, +∞) and ku(t)k

≤ Cku

k

.

– Uniqueness of strong solutions [7, Theorem 3.9]

For any u

∈ V ∩ L

a strong solution with u(0) = u

is unique.

– Smothness [7, Corollary 3.3].

If (u, p) is a strong solution of the Navier-Stokes equation then

∂

∇

u, ∂

∇

p ∈ C((0, T ), L

) ∩ C((0, T ), L

), ∀m, k ≥ 0

and, in particular, u, p ∈ C

( R

× (0, T )) constitute a classical solution of

the Navier-Stokes equations on (0, T ) × R

.

To expand globally the regularity of the Navier-Stokes equation by means of dilation symmetry, the condition (9) must hold.

None of conditions (10), (11), (12) satisfy Corollary 1.

For instance, for n 6= 3 the conditions like (10), (11), (12) would be appro- priate for application of Corollary 1.

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